Skip to main content
Book cover

Mathematical Modelling in Plant Biology pp 107–138Cite as

Modeling Plant Tissue Growth and Cell Division

Abstract

Morphogenesis is the creation of form, a complex process requiring the integration of genetics, mechanics, and geometry. Patterning processes driven by molecular regulatory and signaling networks interact with growth to create organ shape, often in unintuitive ways. Computer simulation modeling is becoming an increasingly important tool to aid our understanding of these complex interactions. In this chapter we introduce computational approaches for studying these processes on spatial, multicellular domains. For some problems, such as the exploration of many patterning processes, simulation can be done on static (non-growing) templates. These can range from abstract idealized cells, such as rectangular or hex grids, to more realistic shapes such as Voronoi regions, or even shapes extracted from bio-imaging data. More dynamic processes like phyllotaxis involve the interaction of growth and patterning, and require the simulation of growing domains. In the simplest case growth can be modeled descriptively, provided as an input to the model. Growth is specified globally, and must be designed carefully to avoid conflicts (growing cells must fit together). We present several methods for this that can be applied to shoots, roots, leaves, and other plant organs. However when shape is an emergent property of the model, different cells or areas of the tissue need to specify their growth locally, and physically-based methods (mechanics) are required to resolve conflicts. Among these are mass-spring, finite element, and Hamiltonian-based approaches.

Keywords

  • Growth Tensor
  • Parametric Coordinates
  • Rest Length
  • Cell Wall Segments
  • Symplastic Growth

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-99070-5_7
  • Chapter length: 32 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   149.00
Price excludes VAT (USA)
  • ISBN: 978-3-319-99070-5
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Hardcover Book
USD   199.99
Price excludes VAT (USA)
Learn about institutional subscriptions
Fig. 7.1
Fig. 7.2
Fig. 7.3
Fig. 7.4
Fig. 7.5
Fig. 7.6
Fig. 7.7
Fig. 7.8
Fig. 7.9
Fig. 7.10
Fig. 7.11
Fig. 7.12
Fig. 7.13
Fig. 7.14
Fig. 7.15
Fig. 7.16
Fig. 7.17
Fig. 7.18
Fig. 7.19
Fig. 7.20
Fig. 7.21

References

  1. Abley K, De Reuille PB, Strutt D, Bangham A, Prusinkiewicz P, Marée AF, Grieneisen VA, Coen E (2013) An intracellular partitioning-based framework for tissue cell polarity in plants and animals. Development 140(10):2061–2074

    CAS  CrossRef  Google Scholar 

  2. Armour WJ, Barton DA, Law AM, Overall RL (2015) Differential growth in periclinal and anticlinal walls during lobe formation in arabidopsis cotyledon pavement cells. Plant Cell 27(9):2484–2500

    CAS  CrossRef  Google Scholar 

  3. Bassel GW, Stamm P, Mosca G, de Reuille PB, Gibbs DJ, Winter R, Janka A, Holdsworth MJ, Smith RS (2014) Mechanical constraints imposed by 3d cellular geometry and arrangement modulate growth patterns in the arabidopsis embryo. Proc Natl Acad Sci 111(23):8685–8690

    CAS  CrossRef  Google Scholar 

  4. Besson S, Dumais J (2011) Universal rule for the symmetric division of plant cells. Proc Natl Acad Sci 108(15):6294–6299

    CAS  CrossRef  Google Scholar 

  5. Bilsborough GD, Runions A, Barkoulas M, Jenkins HW, Hasson A, Galinha C, Laufs P, Hay A, Prusinkiewicz P, Tsiantis M (2011) Model for the regulation of arabidopsis thaliana leaf margin development. Proc Natl Acad Sci 108(8):3424–3429

    CAS  CrossRef  Google Scholar 

  6. Boudon F, Pradal C, Cokelaer T, Prusinkiewicz P, Godin C (2012) L-py: an l-system simulation framework for modeling plant architecture development based on a dynamic language. Front. Plant Sci. 3:76

    CrossRef  Google Scholar 

  7. Boudon F, Chopard J, Ali O, Gilles B, Hamant O, Boudaoud A, Traas J, Godin C (2015) A computational framework for 3d mechanical modeling of plant morphogenesis with cellular resolution. PLoS Comput Biol 11(1):e1003950

    CrossRef  Google Scholar 

  8. Campilho A, Garcia B, Wijk HV, Campilho A, Scheres B et al (2006) Time-lapse analysis of stem-cell divisions in the arabidopsis thaliana root meristem. Plant J 48(4):619–627

    CAS  CrossRef  Google Scholar 

  9. Cieslak M, Runions A, Prusinkiewicz P (2015) Auxin-driven patterning with unidirectional fluxes. J Exp Bot 66:5083–5102. https://doi.org/10.1093/jxb/erv262

    CAS  CrossRef  Google Scholar 

  10. Coen E, Rebocho AB (2016) Resolving conflicts: modeling genetic control of plant morphogenesis. Dev Cell 38(6):579–583

    CAS  CrossRef  Google Scholar 

  11. Coen E, Rolland-Lagan AG, Matthews M, Bangham JA, Prusinkiewicz P (2004) The genetics of geometry. Proc Natl Acad Sci USA 101(14):4728–4735

    CAS  CrossRef  Google Scholar 

  12. de Boer MJ, Fracchia FD, Prusinkiewicz P (1992) A model for cellular development in morphogenetic fields. In: Lindenmayer systems. Springer, Berlin, pp 351–370

    CrossRef  Google Scholar 

  13. de Reuille PB, Routier-Kierzkowska AL, Kierzkowski D, Bassel GW, Schüpbach T, Tauriello G, Bajpai N, Strauss S, Weber A, Kiss A et al (2015) Morphographx: a platform for quantifying morphogenesis in 4d. Elife 4:e05864

    CrossRef  Google Scholar 

  14. De Rybel B, Adibi M, Breda AS, Wendrich JR, Smit ME, Novák O, Yamaguchi N, Yoshida S, Van Isterdael G, Palovaara J et al (2014) Integration of growth and patterning during vascular tissue formation in arabidopsis. Science 345(6197):1255215

    CrossRef  Google Scholar 

  15. Donnelly P, Bonetta D, Tsukaya H, Dengler R, Dengler N (1999) Cell cycling and cell enlargement in developing leaves of Arabidopsis. Dev Biol 215(2):407–419

    CAS  CrossRef  Google Scholar 

  16. Dupuy L, Mackenzie J, Haseloff J (2010) Coordination of plant cell division and expansion in a simple morphogenetic system. Proc Natl Acad Sci 107(6):2711–2716

    CAS  CrossRef  Google Scholar 

  17. el Showk S, Blomster T, Siligato R, Marée AF, Mähönen AP, Grieneisen VA et al (2015) Parsimonious model of vascular patterning links transverse hormone fluxes to lateral root initiation: auxin leads the way, while cytokinin levels out. PLoS Comput Biol 11(10):e1004450

    CrossRef  Google Scholar 

  18. Errera L (1888) Über zellfromen und seifenblasen. Botanisches Centralblatt 34:395–398

    Google Scholar 

  19. Federl P, Prusinkiewicz P (1999) Virtual laboratory: an interactive software environment for computer graphics. In: Computer graphics international, vol 242, pp 93–100

    Google Scholar 

  20. Fernandez R, Das P, Mirabet V, Moscardi E, Traas J, Verdeil JL, Malandain G, Godin C (2010) Imaging plant growth in 4d: robust tissue reconstruction and lineaging at cell resolution. Nat Methods 7(7):547–553

    CAS  CrossRef  Google Scholar 

  21. Feugier FG, Mochizuki A, Iwasa Y. (2005) Self-organization of the vascular system in plant leaves: inter-dependent dynamics of auxin flux and carrier proteins. J Theor Biol 236(4):366–375

    CAS  CrossRef  Google Scholar 

  22. Fukushima K, Fujita H, Yamaguchi T, Kawaguchi M, Tsukaya H, Hasebe M (2015) Oriented cell division shapes carnivorous pitcher leaves of sarracenia purpurea. Nat Commun 6:6450

    CAS  CrossRef  Google Scholar 

  23. Galassi M, Davies J, Theiler J, Gough B, Jungman G, Alken P, Booth M, Rossi F (2002) Gnu Scientific Library. Network Theory Ltd 3

    Google Scholar 

  24. Giavitto JL, Michel O (2001) MGS: a ruled-based language for complex objects and collections. Electron Notes Theor Comput Sci 59(4):1–19

    CrossRef  Google Scholar 

  25. Gierer A, Meinhardt H (1972) A theory of biological pattern formation. Kybernetik 12(1):30–39

    CAS  CrossRef  Google Scholar 

  26. Glazier JA, Graner F (1993) Simulation of the differential adhesion driven rearrangement of biological cells. Phys Rev E 47(3):2128

    CAS  CrossRef  Google Scholar 

  27. Goriely A, Robertson-Tessi M, Tabor M, Vandiver R (2008) Elastic growth models. In: Mathematical modelling of biosystems. Springer, Berlin, pp 1–44

    Google Scholar 

  28. Grieneisen VA, Xu J, Marée AFM, Hogeweg P, Scheres B (2007) Auxin transport is sufficient to generate a maximum and gradient guiding root growth. Nature 449(7165):1008–1013. https://doi.org/10.1038/nature06215. http://dx.doi.org/10.1038/nature06215

    CAS  CrossRef  Google Scholar 

  29. Haselkorn R (1998) How cyanobacteria count to 10. Science 282(5390):891–892

    CAS  CrossRef  Google Scholar 

  30. Heisler MG, Jönsson H (2006) Modeling auxin transport and plant development. J Plant Growth Regul 25:302–312. https://doi.org/10.1007/s00344-006-0066-x

    CAS  CrossRef  Google Scholar 

  31. Hejnowicz Z, Karczewski J (1993) Modeling of meristematic growth of root apices in a natural coordinate system. Am J Bot 80:309–315

    CrossRef  Google Scholar 

  32. Hejnowicz Z, Nakielski J, Hejnowicz K (1984) Modeling of spatial variations of growth within apical domes by means of the growth tensor. ii. Growth specified on dome surface. Acta Soc Bot Pol 53:301–316.

    CrossRef  Google Scholar 

  33. Hervieux N, Dumond M, Sapala A, Routier-Kierzkowska AL, Kierzkowski D, Roeder AH, Smith RS, Boudaoud A, Hamant O (2016) A mechanical feedback restricts sepal growth and shape in arabidopsis. Curr Biol 26(8):1019–1028

    CAS  CrossRef  Google Scholar 

  34. Hofmeister W (1868) Handbuch der physiologishen botanik. Engelmann, Leipzig

    Google Scholar 

  35. Honda H (1978) Description of cellular patterns by Dirichlet domains: the two-dimensional case. J Theor Biol 72:523–543

    CAS  CrossRef  Google Scholar 

  36. Honda H (1983) Geometrical models for cells in tissues. Int Rev Cytol 81:191–248

    CAS  CrossRef  Google Scholar 

  37. Jönsson H, Heisler MG, Shapiro BE, Meyerowitz EM, Mjolsness E (2006) An auxin-driven polarized transport model for phyllotaxis. Proc Natl Acad Sci U S A 103(5):1633–1638. https://doi.org/10.1073/pnas.0509839103. http://dx.doi.org/10.1073/pnas.0509839103

    CrossRef  Google Scholar 

  38. Jönsson H, Gruel J, Krupinski P, Troein C (2012) On evaluating models in computational morphodynamics. Curr Opin Plant Biol 15(1):103–110

    CrossRef  Google Scholar 

  39. Karlebach G, Shamir R (2008) Modelling and analysis of gene regulatory networks. Nat Rev Mol Cell Biol 9(10):770–780

    CAS  CrossRef  Google Scholar 

  40. Kennaway R, Coen E, Green A, Bangham A (2011) Generation of diverse biological forms through combinatorial interactions between tissue polarity and growth. PLoS Comput Biol 7(6):e1002071

    CAS  CrossRef  Google Scholar 

  41. Kierzkowski D, Nakayama N, Routier-Kierzkowska AL, Weber A, Bayer E, Schorderet M, Reinhardt D, Kuhlemeier C, Smith RS (2012) Elastic domains regulate growth and organogenesis in the plant shoot apical meristem. Science 335(6072):1096–1099

    CAS  CrossRef  Google Scholar 

  42. Kramer EM (2008) Computer models of auxin transport: a review and commentary. J Exp Bot 59(1):45–53

    CAS  CrossRef  Google Scholar 

  43. Kramer EM (2009) Auxin-regulated cell polarity: an inside job? Trends Plant Sci 14(5):242–247

    CAS  CrossRef  Google Scholar 

  44. Kuchen EE, Fox S, de Reuille PB, Kennaway R, Bensmihen S, Avondo J, Calder GM, Southam P, Robinson S, Bangham A et al (2012) Generation of leaf shape through early patterns of growth and tissue polarity. Science 335(6072):1092–1096

    CAS  CrossRef  Google Scholar 

  45. Kuhlemeier C (2007) Phyllotaxis. Trends Plant Sci 12(4):143–150

    CAS  CrossRef  Google Scholar 

  46. Kwiatkowska D (2006) Flower primordium formation at the arabidopsis shoot apex: quantitative analysis of surface geometry and growth. J Exp Bot 57(3):571–580

    CAS  CrossRef  Google Scholar 

  47. Lindenmayer A (1968) Mathematical models for cellular interactions in development. I. Filaments with one-sided inputs. J Theor Biol 18(3):280–299

    CAS  CrossRef  Google Scholar 

  48. Lindenmayer A (1968) Mathematical models for cellular interactions in development. II. Simple and branching filaments with two-sided inputs. J Theor Biol 18(3):300–315

    CAS  PubMed  Google Scholar 

  49. Lintilhac PM, Vesecky TB (1984) Stress-induced alignment of division plane in plant tissues grown in vitro. Nature 307(5949):363–364

    CrossRef  Google Scholar 

  50. Lockhart JA (1965) An analysis of irreversible plant cell elongation. J Theor Biol 8(2):264–275

    CAS  CrossRef  Google Scholar 

  51. Louveaux M, Julien JD, Mirabet V, Boudaoud A, Hamant O (2016) Cell division plane orientation based on tensile stress in arabidopsis thaliana. Proc Natl Acad Sci 113(30):E4294–303. https://doi.org/10.1073/pnas.1600677113

    CAS  CrossRef  Google Scholar 

  52. Meinhardt H (1982) Models of biological pattern formation. Academic Press, London

    Google Scholar 

  53. Meinhardt H (2003) Complex pattern formation by a self-destabilization of established patterns: chemotactic orientation and phyllotaxis as examples. C R Biol 326(2):223–237

    CrossRef  Google Scholar 

  54. Merks RM, Guravage M, Inzé D, Beemster GT (2011) Virtualleaf: an open-source framework for cell-based modeling of plant tissue growth and development. Plant Physiol 155(2):656–666

    CAS  CrossRef  Google Scholar 

  55. Mitchison GJ (1980) A model for vein formation in higher plants. Philos Trans R Soc Lond B Biol Sci 207:79–109

    CrossRef  Google Scholar 

  56. Nakielski J (2000) Pattern formation in biology, vision and dynamics, chap. Tensorial model for growth and cell division in the shoot apex. World Scientific, pp. 252–286

    Google Scholar 

  57. Nakielski J, Barlow P (1995) Principal directions of growth and the generation of cell patterns in wild-type and gib-1 mutant roots of tomato (lycopersicon esculentum mill.) grown in vitro. Planta 196(1):30–39

    CAS  CrossRef  Google Scholar 

  58. Nakielski J, Lipowczan M (2013) Spatial and directional variation of growth rates in arabidopsis root apex: A modelling study. PLOS ONE 8(12). https://doi.org/10.1371/journal.pone.0084337. https://doi.org/10.1371/journal.pone.0084337

  59. Neubert MG, Caswell H, Murray J (2002) Transient dynamics and pattern formation: reactivity is necessary for turing instabilities. Math Biosci 175(1):1–11

    CrossRef  Google Scholar 

  60. Prusinkiewicz P, Lane B (2012) Pattern formation in morphogenesis. Springer, Berlin

    Google Scholar 

  61. Prusinkiewicz P, Lane B (2013) Modeling morphogenesis in multicellular structures with cell complexes and l-systems. In: Pattern formation in morphogenesis. Springer, Berlin, pp 137–151

    CrossRef  Google Scholar 

  62. Prusinkiewicz P, Lindenmayer A (1990) Algorithmic beauty of plants. Springer, Berlin

    CrossRef  Google Scholar 

  63. Rebocho AB, Southam P, Kennaway JR, Bangham JA, Coen E (2017) Generation of shape complexity through tissue conflict resolution. eLife 6:e20156

    CrossRef  Google Scholar 

  64. Reinhardt D, Pesce ER, Stieger P, Mandel T, Baltensperger K, Bennett M, Traas J, Friml J, Kuhlemeier C (2003) Regulation of phyllotaxis by polar auxin transport. Nature 426(6964):255–260. https://doi.org/10.1038/nature02081. http://dx.doi.org/10.1038/nature02081

    CAS  CrossRef  Google Scholar 

  65. Rodriguez EK, Hoger A, McCulloch AD (1994) Stress-dependent finite growth in soft elastic tissues. J Biomech 27(4):455–467

    CAS  CrossRef  Google Scholar 

  66. Rolland-Lagan AG, Prusinkiewicz P (2005) Reviewing models of auxin canalization in the context of leaf vein pattern formation in Arabidopsis. Plant J 44(5):854–865. https://doi.org/10.1111/j.1365-313X.2005.02581.x

    CAS  CrossRef  Google Scholar 

  67. Rolland-Lagan AG, Remmler L, Girard-Bock C (2014) Quantifying shape changes and tissue deformation in leaf development. Plant Physiol 165(2):496–505

    CAS  CrossRef  Google Scholar 

  68. Runions A (2008) Modeling biological patterns using the space colonization algorithm. M.Sc. Thesis, University of Calgary

    Google Scholar 

  69. Runions A, Fuhrer M, Lane B, Federl P, Rolland-Lagan AG, Prusinkiewicz P (2005) Modeling and visualization of leaf venation patterns. ACM Trans Graph 24:702–711

    CrossRef  Google Scholar 

  70. Sachs T (1981) The control of patterned differentiation of vascular tissues. Adv Bot Res 9:151–262

    CrossRef  Google Scholar 

  71. Sahlin P, Söderberg B, Jönsson H (2009) Regulated transport as a mechanism for pattern generation: capabilities for phyllotaxis and beyond. J Theor Biol 258(1):60–70

    CAS  CrossRef  Google Scholar 

  72. Sauret-Güeto S, Schiessl K, Bangham A, Sablowski R, Coen E (2013) Jagged controls arabidopsis petal growth and shape by interacting with a divergent polarity field. PLoS Biol 11(4):e1001550

    CrossRef  Google Scholar 

  73. Scarpella E, Francis P, Berleth T (2004) Stage-specific markers define early steps of procambium development in Arabidopsis leaves and correlate termination of vein formation with mesophyll differentiation. Development 131(14):3445–3455

    CAS  CrossRef  Google Scholar 

  74. Scarpella E, Marcos D, Friml J, Berleth T (2006) Control of leaf vascular patterning by polar auxin transport. Genes Dev 20(8):1015–1027. https://doi.org/10.1101/gad.1402406

    CAS  CrossRef  Google Scholar 

  75. Smith R (2011) Modeling plant morphogenesis and growth. New Trends Phys Mech Biol Syst 92:301–336

    CrossRef  Google Scholar 

  76. Smith RS, Bayer EM (2009) Auxin transport-feedback models of patterning in plants. Plant Cell Environ 32(9): 1258–1271. https://doi.org/10.1111/j.1365-3040.2009.01997.x. http://dx.doi.org/10.1111/j.1365-3040.2009.01997.x

    CAS  CrossRef  Google Scholar 

  77. Smith RS, Guyomarc’h S, Mandel T, Reinhardt D, Kuhlemeier C, Prusinkiewicz P (2006) A plausible model of phyllotaxis. Proc Natl Acad Sci U S A 103(5):1301–1306. https://doi.org/10.1073/pnas.0510457103

    CAS  CrossRef  Google Scholar 

  78. Smith RS, Kuhlemeier C, Prusinkiewicz P (2006) Inhibition fields for phyllot actic pattern formation: a simulation study. Can J Bot 84(11):1635–1649

    CrossRef  Google Scholar 

  79. Turing A (1952) The chemical basis of morphogenesis. Philos Trans R Soc Lond B Biol Sci 237:37–52

    CrossRef  Google Scholar 

  80. Wabnik K, Robert HS, Smith RS, Friml J (2013) Modeling framework for the establishment of the apical-basal embryonic axis in plants. Curr Biol 23(24):2513–2518

    CAS  CrossRef  Google Scholar 

  81. Yoshida S, de Reuille PB, Lane B, Bassel GW, Prusinkiewicz P, Smith RS, Weijers D (2014) Genetic control of plant development by overriding a geometric division rule. Dev cell 29(1):75–87

    CAS  CrossRef  Google Scholar 

  82. Žádníková P, Wabnik K, Abuzeineh A, Gallemi M, Van Der Straeten D, Smith RS, Inzé D, Friml J, Prusinkiewicz P, Benková E (2016) A model of differential growth-guided apical hook formation in plants. Plant Cell 28(10):2464–2477

    CrossRef  Google Scholar 

  83. Zienkiewicz OC, Taylor RL (2005) The finite element method for solid and structural mechanics. Butterworth-Heinemann, Boston

    Google Scholar 

Download references

Acknowledgements

Funding is gratefully acknowledged from the Bundesministerium für Bildung und Forschung grants 031A492 and 031A494, the Swiss National Science Foundation SystemsX.ch Plant Growth RTD, Human Frontiers Science Program grant RGP0008/2013 to R.S.S., Marie Skodowska-Curie individual fellowship (Horizon 2020, 703886) to A.R., and the Max Planck Institute for Plant Breeding Research, Cologne, Germany. Some sections of this chapter were adapted from the lecture notes of the Les Houches summer school of 2009 [75]. We would also like to acknowledge Przemyslaw Prusinkiewicz and the members of his lab for helping to formulate many of the ideas appearing in this chapter.

Author information

Authors and Affiliations

  1. Max Planck Institute for Plant Breeding Research, Köln, Germany

    Gabriella Mosca, Milad Adibi, Soeren Strauss, Adam Runions, Aleksandra Sapala & Richard S. Smith

Authors
  1. Gabriella Mosca
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Milad Adibi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Soeren Strauss
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Adam Runions
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Aleksandra Sapala
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Richard S. Smith
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Computational and Systems Biology, John Innes Centre, Norwich, Norfolk, UK

    Prof. Richard J. Morris

Rights and permissions

Reprints and Permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Verify currency and authenticity via CrossMark

Cite this chapter

Mosca, G., Adibi, M., Strauss, S., Runions, A., Sapala, A., Smith, R.S. (2018). Modeling Plant Tissue Growth and Cell Division. In: Morris, R. (eds) Mathematical Modelling in Plant Biology. Springer, Cham. https://doi.org/10.1007/978-3-319-99070-5_7

Download citation